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Now, V is defined by gluing two copies of Y = A1k (0) to each other via the map t → 1/t. Denote the two copies of Y by V1 and V2 . Recalling that Y is affine, isomorphic to Z(x, y) ⊆ A2k , we see that the Vi give an atlas for V as a prevariety. To construct an isomorphism to A1k , consider the regular function defined by 1/(t − 1) on V1 , and by 1/(1/t − 1) = t/(1 − t) on V2 . This defines a morphism to A1k , with inverse morphism defined by sending x to (x + 1)/x in V1 for x = 0, and to x/(x + 1) in V2 for x = −1.

Indeed, this follows immediately from the fact that a regular function on an open subset U of X is the same as a morphism U → A1k . A rational function corresponds to a dominant rational map if and only if it is nonconstant: although it is not immediately obvious what the image of a morphism U → A1k could look like, its closure must be an irreducible closed subset of A1k , and hence is either a single point or all of A1k . 7. The morphism A2k → A2k defined by sending (x, y) to (x, xy) is not an isomorphism, but it is a birational map.

1. Let R be a Noetherian local ring with maximal ideal m, and k = R/m. The Zariski cotangent space of R is defined to be mP /m2P , considered as a k-vector space. If X is an affine algebraic set, and P ∈ X, the Zariski cotangent space of X at P , denoted TP∗ (X), is the Zariski cotangent space of the local ring OP,X . If Noetherian local rings are too abstract, the prototype to have in mind are our local rings OP,X . Note that k has a natural well-defined multiplication on m/m2 , so the first definition makes sense.